\documentclass[12 pt]{article} %sets the font to 12 pt and says this is an article (as opposed to book or other documents)
\usepackage{amsfonts, amssymb} % packages to get the fonts, symbols used in most math
%\usepackage{setspace} % Together with \doublespacing below allows for doublespacing of the document
\oddsidemargin=-0.5cm % These three commands create the margins required for class
\setlength{\textwidth}{6.5in} %
\addtolength{\voffset}{-20pt} %
\addtolength{\headsep}{25pt} %
\pagestyle{myheadings} % tells LaTeX to allow you to enter information in the heading
\markright{Murphy Waggoner\hfill \today \hfill} % put your name instead of Murphy Waggoner
% and put the proposition number from the book
% LaTeX will put your name on the left, the date the paper
% is generated in the middle
% and a page number on the right
\newcommand{\eqn}[0]{\begin{array}{rcl}}%begin an aligned equation - allows for aligning = or inequalities. Always use with $$ $$
\newcommand{\eqnend}[0]{\end{array} } %end the aligned equation
\newcommand{\qed}[0]{$\square$} % make an unfilled square the default for ending a proof
%\doublespacing % Together with the package setspace above allows for doublespacing of the document
\begin{document} % end of preamble and beginning of text that will be printed
% makes the word Proposition and the proposition number bold face
\textbf{Proposition R.231:} % the Proposition number from the book (this one is fictitious)
Prove that $A = \left\{m + n\sqrt{3}\ |\ m,n \in \mathbb{Z} \right\}$ is closed under mulitplication.
% be sure to leave at least one blank line here so that
% the Proof starts with a new paragraph
\textbf{Proof:} % makes the word Proof bold face
Let $A = \left\{m + n\sqrt{3}\ |\ m,n \in \mathbb{Z} \right\},$
and let $m + n\sqrt{3}$ and $p + q\sqrt{3}$ be elements of $A$.
Then
$$\eqn
\left( m + n\sqrt{3} \right)\left(p + q\sqrt{3} \right) & = & mp + mq\sqrt{3} + np\sqrt{3} + 3qn\\
& = & (mp + 3qn) + (mq + np)\sqrt{3}.\\
\eqnend$$
Since $m, n, p, q \in \mathbb{Z}$, $mp + 3nq$ and $ mq + np$ are both integers. Therefore,
$$\left( m + n\sqrt{3} \right)\left(p + q\sqrt{3} \right) \in A,$$
and $A$ is closed under multiplication. \qed
\end{document}
